$p$-adic Integration

Question

I am struggling to compute the following integral \begin{equation} \int_{\mathbb{Z}_p}\exp_p(|x|_p)d\mu \end{equation} where \begin{equation} \exp_p(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} \text{ defined on } |x|_p<p^{-1/(p-1)} \end{equation} I know that in general \begin{equation} \int_{\mathbb{Z}_p}f(|x|_p)d\mu=(1-p^{-1})\sum_{i=0}^{\infty}f(p^{-i})p^{-i} \end{equation} but I am confused by the limited domain of $\exp_p(x)$. Thanks.